A group $H$ is called a retract of a group $G$ if there exist homomorphisms $f:H\longrightarrow G$ and $g:G\longrightarrow H$ such that $g\circ f=id_H$. By a trivial retract, I just mean trivial group.

My question is that:

Is there a group $G$ so that $G$ is retract of every its non-trivial retract?

Thanks in advance.

A group $H$ is called a retract of a group $G$ if there exist homomorphisms $f:H\longrightarrow G$ and $g:G\longrightarrow H$ such that $g\circ f=id_H$. By a trivial retract of $G$, I just mean the trivial group and $G$ itslef.

My question is that:

Is there a group admitting non-trivial retracts, and which is a retract of all its non-trivial retracts"?

Thanks in advance.